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Goldbach's conjecture
Goldbach's conjecture, the strong Goldbach conjecture, says every even number greater than 2 is the sum of two primes; proposed in 1742, it remains unproved.
Goldbach's weak conjecture
Goldbach's weak conjecture, also called the ternary Goldbach conjecture, states every odd integer greater than 5 is a sum of three primes; Helfgott proved it in 2013.
Golden ratio
The golden ratio, or golden section, is the irrational number φ ≈ 1.618, defined by a line divided so the parts keep the same proportion.
Googol
A googol, ten duotrigintillion, is the number 10 to the 100th power, named in 1938 by nine-year-old Milton Sirotta, whose uncle Edward Kasner popularized it.
Gorenstein ring
A Gorenstein ring, named after Daniel Gorenstein, is a commutative Noetherian local ring of finite injective dimension over itself; every Gorenstein ring is Cohen–Macaulay and is its own canonical module.
Graded ring
A graded ring is a ring decomposed as a direct sum of pieces indexed by nonnegative integers, where multiplying elements of degrees i and j gives degree i + j.
Graded vector space
A graded vector space, also called a graded linear space, is a vector space decomposed as a direct sum of subspaces indexed by degree, often integers.
Graham's number
Graham's number is an enormous integer, built from Knuth's up-arrow notation, that arose as a Ramsey theory upper bound and was popularized by Martin Gardner in 1977.
Gram matrix
The Gram matrix is a square matrix whose entries are the pairwise inner products of a set of vectors, encoding lengths, angles, and linear dependence; it is always positive semidefinite.
Graph Cartesian product
The graph Cartesian product, denoted G square H and also called the box product, is a binary operation on graphs whose vertices are ordered pairs, adjacent when they differ in exactly one coordinate.
GraphBLAS
GraphBLAS (Graph Basic Linear Algebra Subprograms) is a specification for an API that expresses graph algorithms as sparse matrix operations over semirings, developed by the open GraphBLAS Forum and modeled on the BLAS standard.
Grassmannian
The Grassmannian, also called the Grassmann manifold, is the space of k-dimensional linear subspaces of a vector space, fundamental to algebraic geometry and named for Hermann Grassmann.
Gray code
Gray code, also known as reflected binary code, is an ordering of binary numbers in which successive values differ in exactly one bit, useful in rotary encoders.
Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search (GIMPS) is a volunteer computing project founded by George Woltman in 1996 that has discovered the last 18 Mersenne primes, including the largest known prime.
Greater-than sign
The greater-than sign (>) is a mathematical symbol indicating the value on its left exceeds the value on its right, first printed in Thomas Harriot's 1631 algebra book.
Greatest common divisor
The greatest common divisor, also known as the highest common factor, is the largest positive integer that divides two integers, found by the Euclidean algorithm from Euclid's Elements.
Green's relations
Green's relations are five equivalence relations defined on a semigroup from its principal ideals, introduced by J. A. Green in 1951 and central to semigroup theory.
Griffiths group
The Griffiths group, named for Phillip A. Griffiths, is the group of homologically trivial codimension-r algebraic cycles on a smooth projective variety modulo algebraically trivial ones.
Gröbner basis
A Gröbner basis (Groebner bases) is a generating set for an ideal in a polynomial ring, introduced by Bruno Buchberger in 1965 and named after his advisor Wolfgang Gröbner.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a tool in homological algebra that computes the derived functors of a composite of two functors; Grothendieck introduced it in 1957.
Grothendieck topos
A Grothendieck topos is a category equivalent to sheaves of sets on a site, introduced by Alexander Grothendieck in SGA 4 (1963–1964) as the framework for étale cohomology.
Grothendieck's Galois theory
Grothendieck's Galois theory is Alexander Grothendieck's axiomatic reformulation of Galois theory, covering space theory, and the algebraic fundamental group, given in SGA 1 (1960/61); it characterizes Galois categories via a fiber functor.
Group (mathematics)
In mathematics, a group is a set with a single operation that is associative and has an identity element and inverses, with examples including integers under addition.
Group action
A group action, also called an action of a group, is a way for a group to operate on a set, defined as a homomorphism into its permutations.
Group algebra of a locally compact group
The group algebra of a locally compact group G is the Banach -algebra L¹(G), integrable functions under convolution, the basic object linking Banach algebra theory to harmonic analysis.
Group cohomology
Group cohomology is a branch of homological algebra that assigns abelian groups Hn(G, A) to a group G, measuring how G acts on a module.
Group extension
A group extension is a group E containing a given group N as a normal subgroup with quotient group Q, usually written as a short exact sequence.
Group homomorphism
A group homomorphism, also called a group morphism or morphism of groups, is a function between two groups that preserves the group operation; its kernel and image underlie the first isomorphism theorem.
Group of Lie type
A group of Lie type is a finite group of rational points of a reductive algebraic group over a finite field, including the classical matrix groups.
Group presentation
A group presentation is a description of a group by a set of generators and relators, written ⟨S | R⟩, giving a compact way to define infinite groups.